"...What about the now carefully articulated and demonstrated 'principles'
such as 'the' Fundamental Theorem of Calculus? At one point did 'it'
become a formulated principle and did those who first exploited the
relationship between derivative and integral somehow or other rely on it?
..."
based on Stacy Langton's basic question:
"Applied to the history of mathematics, however, I think that Unguru's
principle is wrong. I would almost go as far, in fact, as to say that he
has it backward: actually, a mathematical principle can't be invented and
formulated until *after* is has been used."
(a point that was hopefully implied in Robert Tragesser's summary - "I
think it would not be difficult to find many other examples of mathematical
principles which are used before they are explicitly formulated", as
cited by Stacy Langton in a manner that I strongly agree with.)
Archimedes used a form of mod 4 infinite series as the core of his view
of taking slices of geometric shapes, and adding them up, as historians
have titled - the method of exhaustion. Clearly, each slice is seen as an
infinite series, that was restated as a finite series, so that an exact
area could be computed. This history is listed in Dijksterhuis' wonderful
book ARCHIMEDES, as discussed here on HM several times.
However, a point that is often been understated (not by me) is that
Greeks like Archimedes and Eudoxus built their exact aspect of often
used mod 4 series (that can easily be extended to a mod n series)
applying a very old two way form of ancient Egyptian fraction 'logic'.
The oldest principles that were used without being formalized, as this
thread has at times suggested to be a mathematical requirement for
sainthood, come form at least the Egyptian Middle Kingdom, if not the
Old Kingdom. Note that Egyptians and Greek easily wrote between finite
and infinite series, as needed (within the domain of rational numbers),
adding the idea of irrational number, stated as an infinite series, by
dividing by 4, as given by the example:
4A/3 = A + A/4 + A/16 + 1/64 + A/256 + A/(3*256)
where A could be any number, even pi.
To one way of reading history, without the foundational work of Egyptian
proto-number theorists, the eventual 2-way street of differential and
integral calculus, using irrationals and higher order numbers to compute
areas of any shape, may not have developed.
Regards to all,
Milo Gardner